market vs grocery store

market and grocery store both are nouns.

Word	Noun	Adjective	Verb	Adverb
market	Yes	No	Yes	No
grocery store	Yes	No	No	No

As nouns, market and grocery store are synonyms defined as:

Other synonyms of market include food market, grocery.

As nouns, grocery store is a hyponym of market; that is, grocery store is a word with a more specific, narrower meaning than market:

market (noun)	grocery store (noun)
the world of commercial activity where goods and services are bought and sold	a marketplace where groceries are sold
a marketplace where groceries are sold
an area in a town where a public mercantile establishment is set up
the customers for a particular product or service
the securities markets in the aggregate

market (verb)	grocery store (verb)
make commercial
engage in the commercial promotion, sale, or distribution of
deal in a market
buy household supplies